齐次坐标得到非齐次坐标

李群： $$SE(3)$$
$$T\in {\mathbb{R}}^{4\times 4}$$
T = [ R t 0 T 1 ] = [ a b c j d e f k g h i l 0 0 0 1 ] T =

= T=[R0T​t1​]=⎣ ⎡​adg0​beh0​cfi0​jkl1​⎦ ⎤​
一个三维齐次坐标 $P$
P = [ v 1 v 2 v 3 1 ] P =

[v1v2v31]" role="presentation" style="position: relative;">⎡⎣⎢⎢⎢v1v2v31⎤⎦⎥⎥⎥$$\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ 1\end{array}\right]$$

P=⎣ ⎡​v1​v2​v3​1​⎦ ⎤​
T P = [ a v 1 + b v 2 + c v 3 + j d v 1 + e v 2 + f v 3 + k g v 1 + h v 2 + i v 3 + l 1 ] TP=

[av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l1]" role="presentation" style="position: relative;">⎡⎣⎢⎢⎢av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l1⎤⎦⎥⎥⎥$$\left[\begin{array}{c}a{v}_1+b{v}_2+c{v}_3+j\\ d{v}_1+e{v}_2+f{v}_3+k\\ g{v}_1+h{v}_2+i{v}_3+l\\ 1\end{array}\right]$$

TP=⎣ ⎡​av1​+bv2​+cv3​+jdv1​+ev2​+fv3​+kgv1​+hv2​+iv3​+l1​⎦ ⎤​
T P u n _ h o m o g e n e o u s = [ a v 1 + b v 2 + c v 3 + j d v 1 + e v 2 + f v 3 + k g v 1 + h v 2 + i v 3 + l ] TP_{un\_homogeneous}=

[av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l]" role="presentation" style="position: relative;">⎡⎣⎢av1+bv2+cv3+jdv1+ev2+fv3+kgv1+hv2+iv3+l⎤⎦⎥$$\left[\begin{array}{c}a{v}_1+b{v}_2+c{v}_3+j\\ d{v}_1+e{v}_2+f{v}_3+k\\ g{v}_1+h{v}_2+i{v}_3+l\end{array}\right]$$

TPun_homogeneous​=⎣ ⎡​av1​+bv2​+cv3​+jdv1​+ev2​+fv3​+kgv1​+hv2​+iv3​+l​⎦ ⎤​